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We study asymptotic spectral properties of the Bochner-Schr\"odinger operator $H_{p}=\frac 1p\Delta^{L^p\otimes E}+V$ on high tensor powers of a Hermitian line bundle $L$ twisted by a Hermitian vector bundle $E$ on a Riemannian manifold $X$ of bounded geometry under assumption that the curvature form of $L$ is non-degenerate. At an arbitrary point $x_0$ of $X$ the operator $H_p$ can be approximated by a model operator $\mathcal H^{(x_0)}$, which is a Schr\"odinger operator with constant magnetic field. For large $p$, the spectrum of $H_p$ asymptotically coincides, up to order $p^{-1/4}$, with the union of the spectra of the model operators $\mathcal H^{(x_0)}$ over $X$. We show that, if the union of the spectra of $\mathcal H^{(x_0)}$ over the complement of a compact subset of $X$ has a gap, then the spectrum of $H_{p}$ in the gap is discrete and the corresponding eigensections decay exponentially away the compact subset., Comment: 23 pages. arXiv admin note: text overlap with arXiv:2012.14196

Let $F$ be a transversely oriented foliation of codimension 1 on a closed manifold $M$, and let $\phi=\{\phi^t\}$ be a foliated flow on $(M,F)$. Assume the closed orbits of $\phi$ are simple and its preserved leaves are transversely simple. In this case, there are finitely many preserved leaves, which are compact. Let $M^0$ denote their union, $M^1=M\setminus M^0$ and $F^1=F|_{M^1}$. We consider two topological vector spaces, $I(F)$ and $I'(F)$, consisting of the leafwise currents on $M$ that are conormal and dual-conormal to $M^0$, respectively. They become topological complexes with the differential operator $d_{F}$ induced by the de~Rham derivative on the leaves, and they have an $\mathbb{R}$-action $\phi^*=\{\phi^{t\,*}\}$ induced by $\phi$. Let $\bar H^\bullet I(F)$ and $\bar H^\bullet I'(F)$ denote the corresponding leafwise reduced cohomologies, with the induced $\mathbb{R}$-action $\phi^*=\{\phi^{t\,*}\}$. We define some kind of Lefschetz distribution $L_{\text{\rm dis}}(\phi)$ of the actions $\phi^*$ on both $\bar H^\bullet I(F)$ and $\bar H^\bullet I'(F)$, whose value is a distribution on $\mathbb{R}$. Its definition involves several renormalization procedures, the main one being the b-trace of some smoothing b-pseudodifferential operator on the compact manifold with boundary obtained by cutting $M$ along $M^0$. We also prove a trace formula describing $L_{\text{\rm dis}}(\phi)$ in terms of infinitesimal data from the closed orbits and preserved leaves. This solves a conjecture of C.~Deninger involving two leafwise reduced cohomologies instead of a single one., Comment: 176 pages

Given a closed manifold $M$ and a closed regular submanifold $L$, consider the corresponding locally convex space $I=I(M,L)$ of conormal distributions, with its natural topology, and the strong dual $I'=I'(M,L)=I(M,L;\Omega)'$ of the space of conormal densities. It is shown that $I$ is a barreled, ultrabornological, webbed, Montel, acyclic LF-space, and $I'$ is a complete Montel space, which is a projective limit of bornological barreled spaces. In the case of codimension one, similar properties and additional descriptions are proved for the subspace $K\subset I$ of conormal distributions supported in $L$ and for its strong dual $K'$. We construct a locally convex Hausdoff space $J$ and a continuous linear map $I\to J$ such that the sequence $0\to K\to I\to J\to 0$ as well as the transpose sequence $0\to J'\to I'\to K'\to 0$ are short exact sequences in the category of continuous linear maps between locally convex spaces. Finally, it is shown that $I\cap I'=C^\infty(M)$ in the space of distributions. In another publication, these results are applied to prove a Lefschetz trace formula for a simple foliated flow $\phi=\{\phi^t\}$ on a compact foliated manifold $(M,F)$. It describes a Lefschetz distribution $L_{\text{\rm dis}}(\phi)$ defined by the induced action $\phi^*=\{\phi^{t\,*}\}$ on the reduced cohomologies $\bar H^\bullet I(F)$ and $\bar H^\bullet I'(F)$ of the complexes of leafwise currents that are conormal and dual-conormal at the leaves preserved by $\phi$., Comment: 55 pages, index of notation

The paper is devoted to the trace formula for the magnetic Laplacian associated with a magnetic system on a compact manifold. This formula is a natural generalization of the semiclassical Gutzwiller trace formula and reduces to it in the case when the magnetic field form is exact. It differs somewhat from the Guillemin-Uribe trace formula studied in the author's previous work with I.A. Taimanov. Moreover, in contrast to that work, the focus is on the trace formula at the zero energy level, which is a critical energy level. The paper gives an overview of the main notions and results related to the trace formula at the zero energy level, describes various approaches to its proof, and gives concrete examples of its computation. In addition, a brief review of the Gutzwiller trace formula for regular and critical energy levels is given., Comment: 43 pages

The Bochner-Schr\"odinger operator $H_{p}=\frac 1p\Delta^{L^p\otimes E}+V$ on tensor powers $L^p$ of a Hermitian line bundle $L$ twisted by a Hermitian vector bundle $E$ on a Riemannian manifold of bounded geometry is studied. For any function $\varphi\in \mathcal S(\mathbb R)$, we consider the bounded linear operator $\varphi(H_p)$ in $L^2(X,L^p\otimes E)$ defined by the spectral theorem and describe an asymptotic expansion of its smooth Schwartz kernel in a fixed neighborhood of the diagonal in the semiclassical limit $p\to \infty$. In particular, we prove that the trace of the operator $\varphi(H_p)$ admits a complete asymptotic expansion in powers of $p^{-1/2}$ as $p\to \infty$., Comment: 24 pages, v2: a result on asymptotic localization of eigenfunctions is added

We compute the trace formula for the magnetic Laplacian on a compact hyperbolic surface of constant curvature with constant magnetic field for energies above the Mane critical level of the corresponding magnetic geodesic flow. We discuss the asymptotic behavior of the coefficients of the trace formula when the energy approaches the Mane critical level., Comment: 20 pages, v3. Final version

Let $\eta$ be a closed real 1-form on a closed Riemannian $n$-manifold $(M,g)$. Let $d_z$, $\delta_z$ and $\Delta_z$ be the induced Witten's type perturbations of the de~Rham derivative and coderivative and the Laplacian, parametrized by $z=\mu+i\nu\in\mathbb C$ ($\mu,\nu\in\mathbb{R}$, $i=\sqrt{-1}$). Let $\zeta(s,z)$ be the zeta function of $s\in\mathbb{C}$, defined as the meromorphic extension of the function $\zeta(s,z)=\operatorname{Str}({\eta\wedge}\,\delta_z\Delta_z^{-s})$ for $\Re s\gg0$. We prove that $\zeta(s,z)$ is smooth at $s=1$ and establish a formula for $\zeta(1,z)$ in terms of the associated heat semigroup. For a class of Morse forms, $\zeta(1,z)$ converges to some $\mathbf{z}\in\mathbb{R}$ as $\mu\to+\infty$, uniformly on $\nu$. We describe $\mathbf{z}$ in terms of the instantons of an auxiliary Smale gradient-like vector field $X$ and the Mathai-Quillen current on $TM$ defined by $g$. Any real 1-cohomology class has a representative $\eta$ satisfying the hypothesis. If $n$ is even, we can prescribe any real value for $\mathbf{z}$ by perturbing $g$, $\eta$ and $X$, and achieve the same limit as $\mu\to-\infty$. This is used to define and describe certain tempered distributions induced by $g$ and $\eta$. These distributions appear in another publication as contributions from the preserved leaves in a trace formula for simple foliated flows, giving a solution to a problem of C.~Deninger., Comment: 61 pages, 1 figure

We establish the theory of Berezin-Toeplitz quantization on symplectic manifolds of bounded geometry. The quantum space of this quantization is the spectral subspace of the renormalized Bochner Laplacian associated with some interval near zero. We show that this quantization has the correct semiclassical limit., Comment: 14 pages

In this paper, we construct a family of Berezin-Toeplitz type quantizations of a compact symplectic manifold. For this, we choose a Riemannian metric on the manifold such that the associated Bochner Laplacian has the same local model at each point (this is slightly more general than in almost-K\"ahler quantization). Then the spectrum of the Bochner Laplacian on high tensor powers $L^p$ of the prequantum line bundle $L$ asymptotically splits into clusters of size ${\mathcal O}(p^{3/4})$ around the points $p\Lambda$, where $\Lambda$ is an eigenvalue of the model operator (which can be naturally called a Landau level). We develop the Toeplitz operator calculus with the quantum space, which is the eigenspace of the Bochner Laplacian corresponding to the eigebvalues frrom the cluster. We show that it provides a Berezin-Toeplitz quantization. If the cluster corresponds to a Landau level of multiplicity one, we obtain an algebra of Toeplitz operators and a formal star-product. For the lowest Landau level, it recovers the almost K\"ahler quantization., Comment: 21 page

We study the Bochner-Schr\"odinger operator $H_{p}=\frac 1p\Delta^{L^p\otimes E}+V$ on high tensor powers of a positive line bundle $L$ on a symplectic manifold of bounded geometry. First, we give a rough asymptotic description of its spectrum in terms of the spectra of the model operators. This allows us to prove the existence of gaps in the spectrum under some conditions on the curvature of the line bundle. Then we consider the spectral projection of such an operator corresponding to an interval whose extreme points are not in the spectrum and study asymptotic behavior of its kernel. First, we establish the off-diagonal exponential estimate. Then we state a complete asymptotic expansion in a fixed neighborhood of the diagonal., Comment: 30 pages

The article is dedicated to thye memory of a distinguished mathematician Professor Misha Shubin

A quasiclassical approximation is constructed to describe the eigenvalues of the magnetic Laplacian on a compact Riemannian manifold in the case when the magnetic field is not given by an exact 2-form. For this, the multidimensional WKB method in the form of Maslov canonical operator is applied. In this case, the canonical operator takes values in sections of a nontrivial line bundle. The constructed approximation is demonstrated for the Dirac magnetic monopole on the two-dimensional sphere., Comment: 18 pages, v2: minor revisions

We consider the renormalized Bochner Laplacian acting on tensor powers of a positive line bundle on a compact symplectic manifold. We derive an explicit local formula for the spectral density function in terms of coefficients of the Riemannian metric and symplectic form., Comment: v2: several typos corrected, v3: final version

We consider the Bochner Laplacian on high tensor powers of a positive line bundle on a closed symplectic manifold (or, equivalently, the semiclassical magnetic Schr\"odinger operator with the non-degenerate magnetic field). We assume that the operator has discrete wells. The main result of the paper states asymptotic expansions for its low-lying eigenvalues., Comment: 24 pages. arXiv admin note: text overlap with arXiv:1809.06799

We describe transversely oriented foliations of codimension one on closed manifolds that admit simple foliated flows., Comment: 31 pages

A leafwise Hodge decomposition was proved by Sanguiao for Riemannian foliations of bounded geometry. Its proof is explained again in terms of our study of bounded geometry for Riemannian foliations. It is used to associate smoothing operators to foliated flows, and describe their Schwartz kernels. All of this is extended to a leafwise version of the Novikov differential complex., Comment: 46 pages

In this paper, we discuss spectral properties of Laplacians associated with an arbitrary smooth distribution on a compact manifold. First, we give a survey of results on generalized smooth distributions on manifolds, Riemannian structures and associated Laplacians. Then, under the assumption that the singular foliation generated by the distribution is regular, we prove that the Laplacian associated with the distribution defines an unbounded multiplier on the foliation $C^*$-algebra. To this end, we give the construction of a parametrix., Comment: 19 pages. arXiv admin note: text overlap with arXiv:1710.10119, arXiv:1807.06815

The Guillemin-Uribe trace formula is a semiclassical version of the Selberg trace formula and more general Duistermaat-Guillemin formula for elliptic operators on compact manifolds, which reflects the dynamics of magnetic geodesic flows in terms of eigenvalues of a natural differential operator (the magnetic Laplacian) associated with the magnetic field. In this paper, we give a survey of basic notions and results related with the Guillemin-Uribe trace formula and provide concrete examples of its computation for two-dimensional constant curvature surfaces with constant magnetic fields and for the Katok example., Comment: 37 pages, v2: minor corrections, references added

We consider Toeplitz operators associated with the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle on a compact symplectic manifold. We study the asymptotic behavior, in the semiclassical limit, of low-lying eigenvalues and the corresponding eigenfunctions of a self-adjoint Toeplitz operator under assumption that its principal symbol has a non-degenerate minimum with discrete wells. As an application, we prove upper bounds for low-lying eigenvalues of the Bochner-Laplacian in the semiclassical limit., Comment: 36 pages, v3: final version

We show that any generalised smooth distribution on a smooth manifold, possibly of non-constant rank, admits a Riemannian metric. Using such a metric, we attach a Laplace operator to any smooth distribution as such. When the underlying manifold is compact, we show that it is essentially self-adjoint. Viewing this Laplacian in the longitudinal pseudodifferential calculus of the smallest singular foliation which includes the distribution, we prove hypoellipticity., Comment: 39 pages

We study the asymptotic behavior of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle on a symplectic manifold of bounded geometry. First, we establish the off-diagonal exponential estimate for the generalized Bergman kernel. As an application, we obtain the relation between the generalized Bergman kernel on a Galois covering of a compact symplectic manifold and the generalized Bergman kernel on the base. Then we state the full off-diagonal asymptotic expansion of the generalized Bergman kernel, improving the remainder estimate known in the compact case to an exponential decay. Finally, we establish the theory of Berezin-Toeplitz quantization on symplectic orbifolds associated with the renormalized Bochner-Laplacian., Comment: 33 pages, v.2 is a final update to agree with the published paper

In this paper we continue the study of spectral properties of Laplacians associated with an arbitrary smooth distribution on a compact manifold, initiated in a previous paper. Under assumption that the singular foliation generated by the distribution is smooth, we prove that the Laplacian associated with the distribution defines an unbounded regular self-adjoint operator in some Hilbert module over the foliation $C^*$-algebra.

A full off-diagonal asymptotic expansion is established for the generalized Bergman kernels of the renormalized Bochner Laplacians associated with high tensor powers of a positive line bundle over a compact symplectic manifold. As an application, the algebra of Toeplitz operators on the symplectic manifold associated with the renormalized Bochner Laplacian is constructed., Comment: 21 page, v3: final version

Let $M$ be a compact smooth manifold equipped with a positive smooth density $\mu$ and $H$ be a smooth distribution endowed with a fiberwise inner product $g$. We define the Laplacian $\Delta_H$ associated with $(H,\mu,g)$ and prove that it gives rise to an unbounded self-adjoint operator in $L^2(M,\mu)$. Then, assuming that $H$ generates a singular foliation $\mathcal F$, we prove that, for any function $\varphi$ from the Schwartz space $\mathcal S(\mathbb R)$, the operator $\varphi(\Delta_H)$ is a smoothing operator in the scale of longitudinal Sobolev spaces associated with $\mathcal F$. The proofs are based on pseudodifferential calculus on singular foliations developed by Androulidakis and Skandalis and subelliptic estimates for $\Delta_H$., Comment: 21 pages

We study a lattice point counting problem for a class of families of domains in a Euclidean space. This class consists of anisotropically expanding bounded domains, which remain unchanged along some fixed linear subspace and expand in directions, orthogonal to this subspace. We find the leading term in the asymptotics of the number of lattice points in such family of domains and prove remainder estimates in this asymptotics under various conditions on the lattice and the family of domains. As a consequence, we prove an asymptotic formula for the eigenvalue distribution function of the Laplace operator on a flat torus in adiabatic limit determined by a linear foliation with a nontrivial remainder estimate., Comment: 13 pages

We define Seiberg-Witten equations on closed manifolds endowed with a Riemannian foliation of codimension 4. When the foliation is taut, we show compactness of the moduli space under some hypothesis satisfied for instance by closed K-contact manifolds. Furthermore, we prove some vanishing and non-vanishing results and we highlight that the invariants may be used to distinguish different foliations on diffeomorphic manifolds., Comment: To appear in Journal of Geometry and Physics. Final version

This paper deals with semiclassical asymptotics of the three-dimensional magnetic Laplacian in presence of magnetic confinement. Using generic assumptions on the geometry of the confinement, we exhibit three semiclassical scales and their corresponding effective quantum Hamiltonians, by means of three microlocal normal forms \`a la Birkhoff. As a consequence, when the magnetic field admits a unique and non degenerate minimum, we are able to reduce the spectral analysis of the low-lying eigenvalues to a one-dimensional $\hbar$-pseudo-differential operator whose Weyl's symbol admits an asymptotic expansion in powers of $\hbar^{\frac1 2}$.

We revisit the problem of semiclassical spectral asymptotics for a pure magnetic Schr\"odinger operator on a two-dimensional Riemannian manifold. We suppose that the minimal value $b_0$ of the intensity of the magnetic field is strictly positive, and the corresponding minimum is unique and non-degenerate. The purpose is to get the control on the spectrum in an interval $(hb_0, h(b_0 +\gamma_0)]$ for some $\gamma_0>0$ independent of the semiclassical parameter $h$. The previous papers by Helffer-Mohamed and by Helffer-Kordyukov were only treating the ground-state energy or a finite (independent of $h$) number of eigenvalues. Note also that N. Raymond and S. Vu Ngoc have recently developed a different approach of the same problem., Comment: 37 pages

We consider a magnetic Schr\"odinger operator $H^h$, depending on a semiclassical parameter $h>0$, on a compact Riemannian manifold. We assume that there is no electric field. We suppose that the minimal value $b_0$ of the intensity of the magnetic field $b$ is strictly positive. We give a survey of the results on asymptotic behavior of the eigenvalues of the operator $H^h$ in the semiclassical limit., Comment: 20 pages

Continuing the study of bounded geometry for Riemannian foliations, begun by Sanguiao, we introduce a chart-free definition of this concept. Our main theorem states that it is equivalent to a condition involving certain normal foliation charts. For this type of charts, it is also shown that the derivatives of the changes of coordinates are uniformly bounded, and there are nice partitions of unity. Applications to a trace formula for foliated flows will given in a forthcoming paper., Comment: 27 pages

We extend to orbifolds classical results on quantum ergodicity due to Shnirelman, Colin de Verdi\`ere and Zelditch, proving that, for any positive, first-order self-adjoint elliptic pseudodifferential operator P on a compact orbifold X with positive principal symbol p, ergodicity of the Hamiltonian flow of p implies quantum ergodicity for the operator P. We also prove ergodicity of the geodesic flow on a compact Riemannian orbifold of negative sectional curvature., Comment: 14 pages

We investigate the remainder in the asymptotic formula for the number of integer points in a family of bounded domains in the Euclidean space, which remain unchanged along some linear subspace and expand in the directions, orthogonal to this subspace. We prove some estimates for the remainder, imposing additional assumptions on the boundary of the domain. We study the average remainder estimates, where the averages are taken over rotated images of the domain by a subgroup of the group SO(n) of orthogonal transformations of the Euclidean space R^n. Using these results, we improve the remainder estimate in the adiabatic limit formula for the eigenvalue distribution function of the Laplace operator associated with a bundle-like metric on a compact manifold equipped with a Riemannian foliation in the particular case when the foliation is a linear foliation on the torus and the metric is the standard Euclidean metric on the torus., Comment: 13 pages. arXiv admin note: text overlap with arXiv:1006.4977

We consider a magnetic Schr\"odinger operator $H^h=(-ih\nabla-\vec{A})^2$ with the Dirichlet boundary conditions in an open set $\Omega \subset {\mathbb R}^3$, where $h>0$ is a small parameter. We suppose that the minimal value $b_0$ of the module $|\vec{B}|$ of the vector magnetic field $\vec{B}$ is strictly positive, and there exists a unique minimum point of $|\vec{B}|$, which is non-degenerate. The main result of the paper is upper estimates for the low-lying eigenvalues of the operator $H^h$ in the semiclassical limit. We also prove the existence of an arbitrary large number of spectral gaps in the semiclassical limit in the corresponding periodic setting., Comment: 20 pages

We present two versions of the Egorov theorem for orbifolds. The first one is a straightforward extension of the classical theorem for smooth manifolds. The second one considers an orbifold as a singular manifold, the orbit space of a Lie group action, and deals with the corresponding objects in noncommutative geometry.

We continue our study of a magnetic Schr\"odinger operator on a two-dimensional compact Riemannian manifold in the case when the minimal value of the module of the magnetic field is strictly positive. We analyze the case when the magnetic field has degenerate magnetic wells. The main result of the paper is an asymptotics of the groundstate energy of the operator in the semiclassical limit. The upper bounds are improved in the case when we have a localization by a miniwell effect of lowest order. These results are applied to prove the existence of an arbitrary large number of spectral gaps in the semiclassical limit in the corresponding periodic setting., Comment: 33 pages

We discuss asymptotic behavior of the eigenvalue distribution of the differential form Laplacian on a Riemannian foliated manifold when the metric on the ambient manifold is blown up in directions normal to the leaves (in the adiabatic limit). Motivated by analogies with semiclassical spectral asymptotics, we use ideas and notions of noncommutative geometry to suggest a conjectural formula for the eigenvalue distribution in the adiabatic limit, which we call noncommutative Weyl formula. We review known results and discuss the correctness of the noncommutative Weyl formula in each case., Comment: 20 pages

We prove an asymptotic formula for the number of integer points in a family of bounded domains in the Euclidean space with smooth boundary, which remain unchanged along some linear subspace and stretch out in the directions, orthogonal to this subspace. A more precise estimate for the remainder is obtained in the case when the domains are strictly convex. Using these results, we improved the remainder estimate in the adiabatic limit formula (due to the first author) for the eigenvalue distribution function of the Laplace operator associated with a bundle-like metric on a compact manifold equipped with a Riemannian foliation in a particular case when the foliation is a linear foliation on the torus and the metric is the standard Euclidean metric on the torus., Comment: 12 pages

We consider a magnetic Schr\"odinger operator $H^h$, depending on the semiclassical parameter $h>0$, on a two-dimensional Riemannian manifold. We assume that there is no electric field. We suppose that the minimal value $b_0$ of the magnetic field $b$ is strictly positive, and there exists a unique minimum point of $b$, which is non-degenerate. The main result of the paper is a complete asymptotic expansion for the low-lying eigenvalues of the operator $H^h$ in the semiclassical limit. We also apply these results to prove the existence of an arbitrary large number of spectral gaps in the semiclassical limit in the corresponding periodic setting., Comment: 23 pages

First, we survey some results on classical and quantum dynamical systems associated with transverse Dirac operators on Riemannian foliations. Then we illustrate these results by two examples of Riemannian foliations: a foliation given by the fibers of a fibration and a linear foliation on the two-dimensional torus., Comment: 19 pages, to be published in the proceedings of the conference 'Spectrum and Dynamics', Montr\'eal, 2008

First, we review the notion of a Poisson structure on a noncommutative algebra due to Block-Getzler and Xu and introduce a notion of a Hamiltonian vector field on a noncommutative Poisson algebra. Then we describe a Poisson structure on a noncommutative algebra associated with a transversely symplectic foliation and construct a class of Hamiltonian vector fields associated with this Poisson structure., Comment: 13 pages

This paper gives a survey of the index theory of tangentially elliptic and transversally elliptic operators on foliated manifolds as well as of related notions and results in non-commutative geometry., Comment: 116 pages, English translation of the paper published in Russian in Uspekhi Mat. Nauk 64:2(386) (2009), 73-202